MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-mat2pmat Structured version   Visualization version   GIF version

Definition df-mat2pmat 20723
Description: Transformation of a matrix (over a ring) into a matrix over the corresponding polynomial ring. (Contributed by AV, 31-Jul-2019.)
Assertion
Ref Expression
df-mat2pmat matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
Distinct variable group:   𝑚,𝑛,𝑟,𝑥,𝑦

Detailed syntax breakdown of Definition df-mat2pmat
StepHypRef Expression
1 cmat2pmat 20720 . 2 class matToPolyMat
2 vn . . 3 setvar 𝑛
3 vr . . 3 setvar 𝑟
4 cfn 8190 . . 3 class Fin
5 cvv 3389 . . 3 class V
6 vm . . . 4 setvar 𝑚
72cv 1636 . . . . . 6 class 𝑛
83cv 1636 . . . . . 6 class 𝑟
9 cmat 20421 . . . . . 6 class Mat
107, 8, 9co 6872 . . . . 5 class (𝑛 Mat 𝑟)
11 cbs 16066 . . . . 5 class Base
1210, 11cfv 6099 . . . 4 class (Base‘(𝑛 Mat 𝑟))
13 vx . . . . 5 setvar 𝑥
14 vy . . . . 5 setvar 𝑦
1513cv 1636 . . . . . . 7 class 𝑥
1614cv 1636 . . . . . . 7 class 𝑦
176cv 1636 . . . . . . 7 class 𝑚
1815, 16, 17co 6872 . . . . . 6 class (𝑥𝑚𝑦)
19 cpl1 19753 . . . . . . . 8 class Poly1
208, 19cfv 6099 . . . . . . 7 class (Poly1𝑟)
21 cascl 19518 . . . . . . 7 class algSc
2220, 21cfv 6099 . . . . . 6 class (algSc‘(Poly1𝑟))
2318, 22cfv 6099 . . . . 5 class ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦))
2413, 14, 7, 7, 23cmpt2 6874 . . . 4 class (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))
256, 12, 24cmpt 4921 . . 3 class (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦))))
262, 3, 4, 5, 25cmpt2 6874 . 2 class (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
271, 26wceq 1637 1 wff matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
Colors of variables: wff setvar class
This definition is referenced by:  mat2pmatfval  20739
  Copyright terms: Public domain W3C validator